Question: Evaluate the following expression. Your answer must be exact. $(4\sqrt{3}+4i)^3=$
Answer: The Strategy The easiest way to find $z^{n}$ for a complex number $z=({a}+{b}i)$ is using its modulus and argument. Therefore, our solution will consist of the following steps: Find the modulus and argument of $z$. [How is this done, in general?] Find the modulus and argument of $z^{n}$. [How is this done, in general?] Find the rectangular form $z^{n}$. Find the modulus and argument of $(4\sqrt{3}+4i)$ $({4\sqrt{3}}+{4}i)$ is of the form $({a}+{b}i)$, where ${a=4\sqrt{3}}$ and ${b=4}$. Therefore: $\begin{aligned}r&=\sqrt{{a}^2 + {b}^2} \\\\&=\sqrt{({4\sqrt{3}})^2 + ({4})^2} \\\\&=\sqrt{{48}+{16}} \\\\&=8\end{aligned}$ Using the arctangent formula, we have: $\begin{aligned}\theta&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{4}}{{4\sqrt{3}}}\right) \\\\&=30^\circ\end{aligned}$ Since both ${a=4\sqrt{3}}$ and ${b=4}$ are positive, $(4\sqrt{3}+4i)$ lies in Quadrant $1$. Therefore, $\theta$ must be between $0^\circ$ and $90^\circ$, so our answer matches our requirements. Find the modulus and argument of $(4\sqrt{3}+4i)^3$ We found that the modulus and argument of $({4\sqrt{3}}+{4}i)$ are $8$ and $30^\circ$. Therefore, the modulus and argument of $({4\sqrt{3}}+{4}i)^3$ are $8^3=512$ and $(30^\circ)\cdot3=90^\circ$. Find the rectangular form of $(4\sqrt{3}+4i)^3$ Since the argument is $90°$, we know the number lies on the positive side of the imaginary number axis and is therefore a positive pure imaginary number. Since the modulus is $512$, our solution is $512i$. [What does this look like graphically?] [How do we find this algebraically?] Summary $(4\sqrt{3}+4i)^3=512i$